TY - JOUR

T1 - Multigoal-oriented optimal control problems with nonlinear PDE constraints

AU - Endtmayer, B.

AU - Langer, U.

AU - Neitzel, I.

AU - Wick, T.

AU - Wollner, W.

N1 - Funding Information: This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181 ‘Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems’ and the DFG - SPP 1962 ‘Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization’ within the project ‘Optimizing Fracture Propagation Using a Phase-Field Approach’ under the project number . Furthermore the authors thank Fredi Tröltzsch, Huidong Yang and Behzad Azmi for helpful discussions. Additionally we would like to thank the reviewers for their careful readings and suggestions, which improved the paper.

PY - 2020/5/15

Y1 - 2020/5/15

N2 - In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semi-linear monotone PDE and the regularized p-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated with the help of several numerical examples.

AB - In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semi-linear monotone PDE and the regularized p-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated with the help of several numerical examples.

KW - Dual-weighted residuals

KW - Finite elements

KW - Multigoal-oriented a posteriori error estimation

KW - Optimal control

KW - Regularized p-Laplacian

UR - http://www.scopus.com/inward/record.url?scp=85078461211&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1903.02799

DO - 10.48550/arXiv.1903.02799

M3 - Article

AN - SCOPUS:85078461211

VL - 79

SP - 3001

EP - 3026

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 10

ER -